In [1]:
import numpy as np


# softmax¶

• $softmax(x_i) = \frac{e^{x_i}}{\sum_j{e^{x_j}}}$

• \begin{align} (softmax(x + c))_{i}= \frac{e^{x_{i} + c}}{\sum_{j} e^{x_{j} + c}} = \frac{e^{x_{i}} \times e^{c}}{e^{c} \times \sum_{j} e^{x_{j}}} \\ = \frac{e^{x_{i}} \times {e^{c}}}{{e^{c}} \times \sum_{j} e^{x_{j}}} = (softmax(x))_{i} \end{align}

so:

• $softmax(x) = softmax(x + c)$
In [2]:
def softmax(x):
orig_shape = x.shape

if len(x.shape) > 1:
# Matrix
x_max = np.max(x, axis=1).reshape(x.shape[0], 1)
x -= x_max
exp_sum = np.sum(np.exp(x), axis=1).reshape(x.shape[0], 1)
x = np.exp(x) / exp_sum
else:
# Vector
x_max = np.max(x)
x -= x_max
exp_sum = np.sum(np.exp(x))
x = np.exp(x) / exp_sum
#or:  x = (np.exp(x)/sum(np.exp(x)))

assert x.shape == orig_shape
return x

In [3]:
def test_softmax_basic():
"""
Some simple tests to get you started.
Warning: these are not exhaustive.
"""
print("Running basic tests...")
test1 = softmax(np.array([1,2]))
print(test1)
ans1 = np.array([0.26894142,  0.73105858])
assert np.allclose(test1, ans1, rtol=1e-05, atol=1e-06)

test2 = softmax(np.array([[1001,1002],[3,4]]))
print(test2)
ans2 = np.array([
[0.26894142, 0.73105858],
[0.26894142, 0.73105858]])
assert np.allclose(test2, ans2, rtol=1e-05, atol=1e-06)

test3 = softmax(np.array([[-1001,-1002]]))
print(test3)
ans3 = np.array([0.73105858, 0.26894142])
assert np.allclose(test3, ans3, rtol=1e-05, atol=1e-06)

print("You should be able to verify these results by hand!\n")

if __name__ == "__main__":
test_softmax_basic()

Running basic tests...
[0.26894142 0.73105858]
[[0.26894142 0.73105858]
[0.26894142 0.73105858]]
[[0.73105858 0.26894142]]
You should be able to verify these results by hand!


In [4]:
x = np.array([[1,2],[4,3]])

In [5]:
np.max(x, axis=1)

Out[5]:
array([2, 4])
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